Optimal Time-Consistent Investment and Reinsurance Strategy Under Time Delay and Risk Dependent Model

. In this paper, we consider the problem of investment and reinsurance with time delay under the compound Poisson model of two-dimensional dependent claims. Suppose an insurance company controls the claim risk of two kinds of dependent insurance businesses by purchasing proportional reinsurance and invests its wealth in a ﬁnancial market composed of a risk-free asset and a risk asset. The risk asset price process obeys the geometric Brownian motion. By introducing the capital ﬂow related to the historical performance of the insurer, the wealth process described by stochastic delay diﬀerential equation (SDDE) is obtained. The extended HJB equation is obtained by using the stochastic control theory under the framework of game theory. Under the reinsurance expected premium principle, optimal time-consistent investment and reinsurance strategy and the corresponding value function are obtained. Finally, the inﬂuence of model parameters on the optimal strategy is explained by numerical analysis.


Introduction
Since insurance companies have been allowed to enter the financial market for investing risk assets, the optimal investment strategy has become an important research topic in recent years. Many literature have studied the maximization of the utility of the terminal wealth or the minimization of the ruin probability of the insurer. Browne [1] uses the surplus process given by the diffusion risk model to study the investment problem of maximizing the utility of the terminal wealth and minimizing the ruin probability of an enterprise and obtains the explicit optimal solution. Hipp and Plum [2] apply the Cramer-Lundberg model to describe the insurance surplus process, based on the assumption that there is only one risky asset in the financial market and the time is discrete; the investment problem is studied. Wang et al. [3] use martingale approach to study the optimal portfolio selection of insurers under the criteria of mean-variance and constant absolute risk aversion utility maximization. For more similar literature, see Liu and Yang [4], Yang and Zhang [5], Wang [3], and Bai and Guo [6].
In addition to market risk, the insurer will also consider insurance risk. It is impossible to avoid insurance risk by investing in bonds and other assets in the market alone. However, reinsurance business provides a way for the insurer to avoid this risk. In recent years, this approach has been widely concerned. Reinsurance business mainly adopts two different forms of insurance: excess-of-loss reinsurance and proportional reinsurance. Promislow and Young [7] first investigate the proportional reinsurance and investment. Bauerle [8] considers proportional reinsurance and investment also, and the optimal explicit solution of the investment-reinsurance problem is obtained under the mean-variance criterion. Zeng and Li [9] also study proportional reinsurance and obtain the efficient frontier of the mean-variance under the multidimensional risky asset model. e stock price in the above model generally follows the geometric Brownian motion; the market price of stock-related risk is constant, but in the real market, stock price may have other characteristics, such as stochastic volatility. Liang et al. [10] used the Ornstein-Uhlenbeck process to characterize the instantaneous return of stocks and obtained the optimal reinsurance and investment strategy. Gu et al. [11] investigate the excess-ofloss reinsurance-investment problem under the constant elasticity variance (CEV) model. ere are two deficiencies in the above literature that deserve further discussion. On the one hand, these literature implicitly assume that all insurance businesses of insurers are independent of each other, so they only study the investment and reinsurance of a single insurance business. However, in the real insurance market, there are often interdependencies between insurance businesses. For example, during the 2019-nCoV, medical claims and death claims often occur together. In order to depict this kind of dependency between different insurance businesses, the risk dependent model is proposed. e main works in this area are as follows. Yuen et al. [12], taking the expected utility maximization of the terminal wealth as the criterion, considered the optimal proportional reinsurance problem with multidimensional risk dependence by using the diffusion approach method. For the detailed process of diffusion approximate to the compound Poisson process, see Gandell [13]. Liang and Yuen [14], under the principle of variance premium, investigated the optimal proportional reinsurance of the Poisson model and diffusion approximation model. Ming et al. [15] derive the explicit expression of the optimal proportional reinsurance under the mean-variance criterion by using stochastic linear quadratic control. Considering the combination of investment and reinsurance, Bi et al. [16] obtains the optimal investment-reinsurance strategy for mean-variance under the diffusion approximation model. Bi and Chen [17], under the criterion of maximizing the expected utility of terminal wealth, arrived at the optimal investment and reinsurance strategies. On the other hand, most of the literature on optimal investment-reinsurance and other optimal control problems focus on time-delay free controlled systems. In fact, financial markets tend to rely on the past, Chang [18] considers the investment and consumption problems related to the return on risk assets and the historical performance. Federico [19] introduces the time-delay state process by considering the capital inflow/ outflow related to performance. Peng et al. [20] and Yu et al. [21] study the optimal dividend policy based on observing the information of past time points to determine the behavior of the next moment. In fact, this is a discrete case of time delay. However, the stochastic control problems of systems with time-delay state may be infinite-dimensional in continuous cases; hence, it is difficult to find the analytical solutions. Only in some special cases, it is finite-dimensional and the problem has explicit solution. Elsanosi et al. [22], ∅ksendal and Sulem [23], and David [24] provide a theoretical basis for solving such problems. Shen and Zeng [25] first introduced the time delay in the investment and insurance problem. ey introduced the inflow/outflow of capital in the wealth process of insurer and then depicted the wealth process of insurance companies through the stochastic delay differential equation (SDDE). After that Li [26] and Lai [27] studied the optimal investment-reinsurance problem with time delay under Heston and CEV models, respectively. Inspired by the above research, this paper combines risk dependence with time delay to consider investment-reinsurance problem. e structure of the rest of this paper is as follows. In Section 2, the financial model framework of this paper is given, assuming that an insurer can invest in a riskfree asset and a risky asset, and in the case of two-dimensional dependent claim compound Poisson model and the introduction of the historical performance of the insurance company, the company's wealth process with time delay is obtained. In Section 3, considering the mean-variance preference criterion, the time-inconsistent optimization problem is defined, and the extended HJB equation is obtained by using the stochastic control theory in the framework of game theory. In Section 4, under the principle of reinsurance expected premium, the explicit solutions of optimal investment and reinsurance strategies and their corresponding value functions are derived. In Section 5, the numerical calculation process of optimal investment and reinsurance strategies are introduced through numerical examples, and the influence of important model parameters on optimal strategy is analyzed. Section 6 concludes this paper.

The Model
Suppose that model is based on the probability space T] , P) of information flow which satisfies the general assumptions of right continuity and completeness, where T is a finite constant, representing the operation cycle of an insurance company, and F { } t∈[0,T] is the sum of information available up to time t. All stochastic processes involved in this paper are assumed to adapt to F { } t∈[0,T] . Suppose an insurer has an insurance portfolio business, which is composed of two different insurance businesses, such as medical insurance and death insurance. Suppose that the random variable Y i , i ≥ 1 represents the claim amount of the first type of insurance business; they are independent and have the same distribution function F Y (y). Z i , i ≥ 1 represents the claim amount of the second type of insurance business; they are independent and have the same distribution function F Z (z). We assume that if y ≤ 0, then F Y (y) � 0. Otherwise, 0 < F Y (y) ≤ 1. And also assume that if z ≤ 0, then F Z (z) � 0. Otherwise, 0 < F Z (z) ≤ 1. In addition, their moment generating functions M Y (ι) and M Z (ι) exist. e cumulative claim process of the two insurance businesses are as follows: where N 1 (t) and N 2 (t) represent the number of claims for the first and second categories of insurance business up to time t, respectively. And suppose Y i , i ≥ 1 , Z i , i ≥ 1 , N 1 (t) t > 0 , and N 2 (t) t > 0 are independent of each other.
For different insurance businesses, it is assumed that they are interdependent as follows: 2 Mathematical Problems in Engineering where N(t), N 1 (t), and N 2 (t) are three independent Poisson processes and the corresponding intensities are λ, λ 1 , and λ 2 , respectively. erefore, the total claim amount of these two types of the insurance business is Suppose for arbitrary ι ∈ (0, ζ), E[Ye ιY ] and E(Ze ιZ ) exist. And, for some ζ ∈ (0, ∞], there are For convenience of writing, we define Considering the financial market, it is assumed that assets are traded continuously in time interval [0, T], and tax and transaction costs are not considered. Suppose the insurer can invest its wealth in the financial market composed of a risk-free asset and a risky asset. e risk-free asset price process e risky asset price process S(t) { } is as follows: where r, α( > r), and σ( > 0) are constants, representing riskfree interest rate, drift rate, and volatility, respectively. Define α ≔ α 1 − r.
As usual, the surplus process from the insurer up to time t is defined as follows: where R 0 is the initial surplus and c is the premium rate. In addition, it is assumed that insurance companies can continuously reinsurance insurance business in a certain proportion to control business risk. We denote the retention ratio of categories 1 and 2 insurance business by q 1 (t) ∈ [0, 1] and q 2 (t) ∈ [0, 1]. When the claim occurs, the insurance company pays q 1 (t)Y i or q 2 (t)Z i , while the reinsurance company pays (1 − q 1 (t))Y i or (1 − q 2 (t))Z i . Let the reinsurance rate be δ(q 1 (t), q 2 (t)) at time t. Let X(t) denote the wealth process of insurance companies at time t, p 1 (t) denote the amount of capital invested in the risky asset, and then X(t) − p 1 (t) denote the amount of wealth invested in the risk-free asset. e investment-reinsurance strategy π(t) ≔ (p 1 (t), q 1 (t), q 2 (t)) will be applied by the insurer. Given an investment-reinsurance strategy π(t), the wealth process X π (t) { } of an insurer satisfies the following stochastic differential equation: Next, we consider the influence of historical performance on the wealth process. Suppose that f(t, X π (t) − L π (t), X π (t) − M π (t)) represents the inflow/outflow of capital, then the wealth process of insurers with time delay is given by the following stochastic delay differential equation (SDDE): To make the problem easier to deal with, consider a linear capital inflow/outflow function, that is, where c 1 > 0 and c 2 > 0 are constants, − h e Au du, and M π (t) � X π (t − h) represent the integrated, average, and point by point delay information of wealth process in time interval [t − h, t]. A( ≥ 0) and h( ≥ 0) are given average parameters and delay parameters, respectively. Note that L π (t) is defined as the weighted average value of wealth process X π (·) in time interval [t − h, t], and the exponential decay factor e Au (u ∈ [− h, 0]) represents the weight. When h � 1, X π (t) − L π (t) and X π (t) − M π (t) represent the average gain or loss and absolute gain or loss of wealth of insurers in the last operating cycle. Because the inflow/ outflow of capital is closely related to the past performance of the wealth process. If the past performance is good, the company will give part of its earnings to shareholders or give bonuses to the management, which shows the outflow of capital, i.e., f > 0. At this time, X π (t) > L π (t) and X π (t) > M π (t). On the contrary, if the past performance of the insurance company is not good, the company needs additional financing to achieve the predetermined goal. is shows capital inflow, i.e., f < 0 when X π (t) < L π (t) and X π (t) < M π (t). erefore, the function f(·, ·, ·) considers the average and absolute performance of the wealth process Mathematical Problems in Engineering Substituting (10) into (9), the following stochastic delay differential equation (SDDE) is obtained: , which can be interpreted as that the insurance company has the initial wealth of x 0 at − h. ere is no business operation during [− h, 0], and the wealth has no change. e integrated delay wealth initial value can be calculated to get .

Optimization Problem
To take historical operating performance into account, the insurer will focus on both terminal wealth X π (T) and historical average operating performance L π (T); thus, the following objective function is defined: where risk aversion coefficient ω( > 0) and delay parameter β(∈ [0, 1]) are constants. E t,x,l,m [·] and Var t,x,l,m [·] represent conditional expectation and conditional variance based on is the weight of L π (t), indicating the degree of terminal wealth affected by historical average performance. If we write In addition, according to Chang [18], delay optimal control problem is generally an infinite-dimensional problem. In order to obtain the optimal solution, some additional conditions will be attached. We assume that the value function V(·) is only related to x and l, but L π (t) is related to M π (t); in order to make V(·) only depend on (t, x, l), the problem can obtain the optimal solution, and we assume the following conditions hold: erefore, this paper aims at the following optimization problems: where

Remark 1
(i) According to Shen and Zeng [25], condition (13) can be regarded as exogenous technical conditions that need to be determined in advance by the insurance company. Firstly, the average delay wealth L π (t) and point by point delay wealth M π (t) are determined by selecting the average parameter A and delay time h. Secondly, it selects the weight β. Finally, it calculates the weight ratios and X π (t) − M π (t) according to the two assumptions in (13) and adjusts the inflow/ outflow of capital accordingly. (ii) Because there is a nonlinear function of the expectation of the terminal value wealth in the variance term, problem (14) is time inconsistent, which leads to the failure of Bellman's optimal principle. Many works of literature deal with the mean-variance problem by setting a precommitment, so the optimal strategy obtained are time-inconsistent. However, for a rational decision maker, time consistency is often not negligible. Rational decision makers hope that the equilibrium strategy they find is not only optimal at this time but also optimal in the future with the evolution of time, that is to say, the equilibrium strategy is time consistent. erefore, for problem (14), this paper aims to find the equilibrium strategy. 4 Mathematical Problems in Engineering Choose arbitrarily π ∈ Π, t > 0, and ε > 0 and define the control law π ε : We call that π is an equilibrium strategy if lim ε↓0 inf(J(t, x, l; π) − J(t, x, l; π ε )/ε) ≥ 0 for any t and π. If the equilibrium strategy π exists, the equilibrium value function is defined as V(t, x, l) � J(t, x, l; π).
According to Definition 2, the equilibrium strategy is time consistent. For simplicity, we denote that is twice continuously differentiable on R, and ϕ(·, ·, l) is once continuously differentiable on R. To provide verification theorem and derive conveniently extended HJB equation, for , and given control law π, we define variational operator as follows: e following theorem provides verification for the extended HJB equation in problem (14). (14), we assume that there exist two real-valued functions

Theorem 1 (verification theorem). For problem
x, l), and π is an equilibrium investment-reinsurance strategy.
e proof process of eorem 1 is similar to that of Björk et al. [28], so it is omitted here.
In Definition 1, the policy set Π is allowed to require the reinsurance policy to satisfy the constraint q 1 (t) ∈ [0, 1] and q 2 (t) ∈ [0, 1]. To facilitate the solution, we do not consider this constraint temporarily and record all the policy sets satisfying (i) and (iii) as Π. According to the variational Mathematical Problems in Engineering 5 operator (16), the extended HJB (17) can be expanded as follows: Suppose that the solution of the above extended HJB equation has the following structure: with the boundary condition H 1 (T) � H 2 (T) � 1 and Differentiating V and g with respect to t, x, and l, we obtain rough simple calculation, we can also obtain where According to c 2 � βe − Ah , we have For the convenience of writing, let

Optimal Time-Consistent Strategy
is section assumes that the reinsurance premium rate is calculated by the expected premium principle, i.e., where η 1 and η 2 are the reinsurer's safety loading of the insurance business.
Substituting the above formula into (24), we have To facilitate derivation, we rewrite (25) as Differentiating h(p, q 1 , q 2 ) with respect to p 1 , q 1 , and q 2 , we can derive From (29), we obtain the following Hessian matrix: Mathematical Problems in Engineering where B � (28) is concave with respect to (p 1 , q 1 , q 2 ).
Proof. In order to prove Lemma 1, we only need to prove that the Hessian matrix is negative definite. From (43), we know H 2 (t) ≠ 0, thus H 2 2 (t) > 0. According to (30), we only need to prove that matrix B is positive definite.
∀C � (c 1 , c 2 , c 3 ) ∈ R 3 and C ≠ 0. Let (·) tr denote the transposition of a vector or matrix, then So, matrix B is positive definite. From (29), we have By solving the above equations, we can obtain . From Lemma 1, we know that ( p 1 (t), q 1 (t), q 2 (t)) is the point where function h(p 1 , q 1 , q 2 ) takes the maximum value. Putting ( p 1 (t), q 1 (t), q 2 (t)) into (22), we can obtain According to we have By separating variables of (x + βl), we can obtain By solving the above equations, we have According to the above discussion, the following proposition can be obtained.

Proposition 1.
For problem (14), the time-consistent investment-reinsurance strategy in set Π is as follows: e corresponding equilibrium function is where H and F are given by (43).
From Lemma 2, we will investigate the optimal results in the following four cases: Next, the optimal time-consistent strategy π * (t) � (p * 1 (t), q * 1 (t), q * 2 (t)) in admissible strategy set Π and the corresponding value function V(t, x, l) are discussed. In order to have a clear classification discussion, it is assumed that r − c 1 − c 2 + β ≥ 0.
It is easy to see that t 6 ≤ t 1 ≤ t 7 .

Remark 2. (i) Since
V(t, x, l) is a continuous function for any (t, x, l) ∈ [0, T] × R × R. Furthermore, which includes that V(t, x, l) is a classical solution to the extended HJB (18). (ii) According to eorem 2, the investment and reinsurance strategy of the insurer is not directly affected by the average parameter A and the delay time h, but according to (13), the average parameter A and the delay time h have an indirect influence on the investment and reinsurance strategy of insurance companies. (iii) Note that, in the classification discussion of eorem 2, in order to make the classification clear, we assume that r − c 1 − c 2 + β ≥ 0. For r − c 1 − c 2 + β < 0, we can also make a similar discussion. (14) degenerates to the case without time delay. Corollary 1. Without time delay, the optimal time-consistent investment and reinsurance policies of problem (14) are as follows: (i) If Case 1 holds, the optimal investment-reinsurance strategies for problem (14) are and the value function is given by (ii) If Case 2 holds, the optimal investment-reinsurance strategies for problem (14) are and the value function is given by (iii) If Case 3 holds, the optimal investment-reinsurance strategies for problem (14) are and the value function is given by (iv) If Case 4 holds, the optimal investment-reinsurance strategies for problem (14) are and the value function is given by

Numerical Simulations
In this section, Example 1 will be used to illustrate the specific numerical calculation process of finding the optimal time-consistent strategy, and Example 2 will be used to analyze the influence of important parameters on the optimal time-consistent strategy. Assuming that the claim amount Y i and Z i are exponentially distributed with parameters ξ 1 and ξ 2 , respectively, then μ 1Y � 1/ξ 1 , μ 1Z � 1/ξ 2 , b 1 � 2(λ + λ 1 )/ξ 2 1 , and b 2 � 2(λ + λ 2 )/ξ 2 2 .
Example 2. If there is no special description in this example, the basic parameter values are as follows: η 1 � η 2 � 0.7, ξ 1 � 2, ξ 2 � 3, λ � 3, λ 1 � 2, λ 2 � 4, α 1 � 0.5, σ � 0.2, r � 0.18, A � 0.1, β � 0.1, h � 0.2, and ω � 0.5. Figures 1 and 2 depict the influence of risk aversion parameter ω and delay parameter β on the optimal timeconsistent investment strategy. From Figure 1, we can see that the optimal time-consistent investment strategy p 1 (t) decreases with the increase of risk aversion parameter ω, that is to say, the higher the risk aversion degree of the insurer is, the less the amount of risk investment will be. Because parameter β includes the information of average parameter A and delay h, it is a comprehensive time-delay parameter, so we only analyze β. Figure 2 shows that the larger the delay parameter β is, the larger the number of investment in risky assets will be. Note that if β � 0, then the insurer decisionmaking is only based on the current information, so it may take short-term risk-taking behavior for the immediate possible high return. For β > 0, when the insurer is making decision, the comprehensive performance in the past period will be taken into account. Insurer focuses on information in a period when making decisions. According to (12), the greater the value of β, the greater the proportion of average   wealth in performance measurement. at is, the insurer can change the inflow/outflow of the insurer's capital by adjusting the size of the parameter beta, thus changing the risk faced by the insurer. e bigger the beta, the smaller the risk, so the insurer will consider increasing the number of risky assets. Figures 3-6 depict the influence of risk aversion coefficient ω and delay parameter β on two types of insurance reinsurance. According to Figures 3 and 4, q 1 (t) and q 2 (t) decrease with respect to ω. e higher the risk aversion degree of the insurer, the more reinsurance he will buy to reduce his risk, so the retention ratio of q 1 (t) and q 2 (t) will be reduced. Figures 5 and 6 show that the retention ratio q 1 (t) (q 1 (t)) increase with respect to the parameter β. As the impact of β on investment strategy p 1 . e larger the β, the stronger the insurer's ability to adjust capital inflow/outflow, that is, the stronger the insurer's risk control ability. To a certain extent, the profitability of the insurer will be stronger, so the insurer will reduce the purchase of reinsurance, and the proportion of reinsurance retention q 1 (t) (q 1 (t)) will increase. is is consistent with economic reality, which the more information investors observe, the more profit they will make. Figures 7-9 depict the effect of the claim intensity λ 1 , λ 2 , and λ on reinsurance. In Figure 7, the larger the λ 1 is, the larger the q 1 (t) is and the smaller the q 2 (t) is. Because the      larger the λ 1 is, the greater the expected claim amount of the first type of insurance business will be, so the insurer will purchase more reinsurance for the first type of insurance business and reduce the proportion of retained insurance q 1 (t). At this time, λ 2 will remain unchanged, that is, the expected claim amount of the second type of insurance business will remain unchanged. Based on the consideration of constant total risk and more profits, the insurer will increase the retention ratio q 2 (t) of reinsurance. A similar analysis can explain why; with the increase of λ 2 , q 1 (t) decreases and q 2 (t) increases in Figures 8 and 9 which shows that the retention ratios q 1 (t) and q 2 (t) of the two types of insurance businesses decrease with the increase of lambda. Because the larger the lambda is, the greater the expected claim amount of the two types of insurance businesses will be. erefore, in order to control the risk within a certain range, the insurer will buy more reinsurance for the two types of insurance businesses and reduce the retention ratio q 1 (t) and q 2 (t).

Conclusion
In this paper, we study the optimal investment-reinsurance problem with delay and risk dependence under the meanvariance preference criterion. Considering the time-delay effect and risk dependence, we obtain the extended HJB equation with delay based on the time delay stochastic control framework and the equilibrium stochastic control method. e results show that the optimal time-consistent investment and reinsurance strategy will be affected by the time delay effect. e larger the capital flow related to the historical business performance, the greater the risk faced by the insurance company. In a prudent attitude, the insurer will reduce the amount invested in a risk asset and reduce the reinsurance retention ratio of all insurance businesses. In addition, risk dependence is linked by common risk shock sources. e greater the risk common shock intensity is, the smaller the reinsurance retention ratio will be. From the numerical analysis results, we can see not only the numerical calculation process of the optimal strategy but also the intuitive verification of the above conclusions.
In this paper, we study the risk assets under geometric Brownian motion. To better simulate the real financial market, the following research will consider the introduction of CEV, Heston, and other stochastic volatility models, Vasicek, CIR, and other stochastic interest rate models.
Data Availability e data in this paper can be used publicly.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 19